@wolftune

The above examples illustrate two issues. Keep in mind that the goal of some supporters of cardinal PR is not to satisfy a strong form of Proportionality for Solid Coalitions (PSC); but to elect one moderate centrist, and a representative for each solid coalition with a quota of supporters. This should often be an achievable goal as the number of candidates entitled to a seat under PSC is usually less than the number of seats to be filled. But by starting off with awarding a seat to the overall score winner electing a moderate is prioritized over PSC.

A second issue, seen in the second example, is that a candidate receiving only low scores can beat a candidate that is clearly entitled to a seat under PSC. This is due to a downward shift in what constitutes a neutral score as the number of seats increases. What I mean by neutral score is the minimum score that a candidate must exceed to be entitled to a seat if everyone bullet votes. This would be the corresponding score to the vote exceeded in the Droop Quota. For 0 to 5 score this would be 2.5 for a single winner election, 1 ⅔ for two winners, 1.25 for three, 1 for four, etc. For a six seat election in the above example a score of 1 isn't a low score of disapproval it's a modest score of approval.

What could be done about these issues?

1: Educate voters on strategy.
Allocated Score will achieve a proportional outcome, if all voters engage in bullet voting. Rating all candidates that the voter doesn't approve of with a score of zero is essential. The ballot options could be changed from (0, 1, 2, 3, 4, 5) to (0, 0, 0, 3, 4, 5) to stress the point. The description of STAR-PR doesn't have the top-two runoff step of STAR so there's no incentive to use intermediate scores as it is.

2: Award seats to candidates that qualify for seats based on their top scores alone first. For example: count only the ratings of 5 cast then seat the candidates that have a score of 5 times the Droop quota, following a procedure similar to the first round of PR-STV. Afterwards this would switch to Allocated Score instead of the elimination round. This should prioritize proportionality, while preserving some of the attributes of Allocated Score. Intermediary rounds could be included, counting scores of four and five and a round counting scores of three, and five in a Bucklin like manner. This would ensure that even parties that split their vote between multiple candidates would get priority over an overall score winner. There would be a confirmation process to handle surplus votes and make sure votes aren't counted twice, that would seat individual candidates sequentially, but this would fundamentally change the process from a series of single winner elections to rounds of multi-winner elections. Multiple candidates could meet the Droop quota to win the round, but the ballot deweighting process could result in not all winners being confirmed.

3: Transform the scores to adjust for the shift in the neutral point.
By adjusting the scores before they are tallied voters can score on the more familiar scale used for single winner elections and have them corrected for the distortion caused by having multiple winners.
Example: four winner election 0 to 5 ballot. The neutral score would be 1. The balance test for the equal vote criteria also changes for multi-winner. Instead of the familiar one vote canceling one vote, four votes in opposition are necessary to cancel each vote in favor(see balance test below). A score of zero remains zero, 1 becomes ⅓, 2 becomes ⅔, 3 becomes 2 ⅔, 4 becomes 3 ⅔, and 5 remains 5. Consider if four voters give a candidate a zero and one a five. Total score will be 5 and the average score will be 1. If four give a 1 and one a 4. Total score would be 4 * ⅓ + 3 ⅔ = 5 and an average of 1. If four give a score of 2 and one a 3. Total score would be 4 * ⅔ + 2 ⅓ = 5 and an average of 1. With these adjustments in place the potential harm to a preferred candidate of giving a lesser candidate an intermediate score will be on par with single winner systems.

Balance Test for Score (Based on Droop Quota rather than majority, to be applicable to any number of winners) : There shall be a set of ballots of size one plus the number of seats to be filled, such that if one of the ballots is cast any way possible the other ballots can be cast to cancel out the vote leaving the average scores for each candidate unchanged. The ballots shall be scored such that, scores of approval on one of the ballots shall be countered by the opposing score of disapproval on the other ballots, and a score of disapproval shall be countered by the opposing score of approval on one of the other ballots and the score of disapproval shall be shared with the remaining ballots.

4:Only count a fraction of the votes for each candidate.
Sequential Monroe avoids the issues by only counting the top Hare quota of ballots for each candidate. With the vast majority of low scores left uncounted, the results are largely unaffected by the distortion of scores in multi-winner elections. However, with little more than the candidates core supporters contributing, the results will lean less moderate until the final round.

What if STAR-PR was not a misleading name; and included an automatic top-two runoff based on all ballot preferences before each selection was finalized?

With the current version of Allocated Score if all voters make use of intermediate scores it will destroy any semblance of proportionality. If all voters resort to min-max voting to protect proportionality the runoff will be irrelevant. More likely voters most interested in getting a particular candidate or party elected or dead set against a particular candidate, will resort to min-max voting while voters torn between objectives will make use of the full range of scores. Vote splitting among min-max voters could result in parties failing to get a seat, and the voters making full use of all scores will have an outsized influence in getting their lesser evils elected.

Score based PR systems have the potential to provide a balance between different interests. With some of the above changes in place, parties with a quota of supporters should get one of their top two candidates elected, even if they split their votes between multiple candidates, but could end up with the one preferred by voters overall, rather than the one preferred by party members.

@sarawolk did you check spam folders? When I set up an account, Gmail auto-filed the confirmation email as spam.

@sander, sequential bottom two runoff, is a Smith efficient Condorcet procedure. It will always select a member of the Smith set and the Condorcet winner if there is one. Though the procedure you describe is different, it will produce the same results as Smith//Score. With score relegated to the relatively trivial role of cycle breaking, properties of Condorcet systems will predominate. By contrast the top two runoff of STAR will block a Condorcet loser, but a Condorcet winner can still lose if they don't get a good enough score to get in the runoff. This is enough for STAR to retain some significant properties of Score.

@toby-pereira I came up with 67 seats for A and 33 seats for B, but that was based on the example in the Wikipedia wiki not Electowiki. That example suggests a cumulative cost tally for each party that is added on to after each seat is apportioned. Party A would have a starting cost of 1/3 or 2/6 and B would be 1/2 or 3/6. When A is selected 1/3 or 2/6 is added to A's cost and (1/2)/3 or 1/6 is added to B's cost to reflect the fraction of B's voters who also voted for A. When B is selected 1/2 or 3/6 is added to B's cost and (1/3)/2 or 1/6 is added to A's. Each round goes to the candidate with the lowest cumulative cost. So the first round would go to A, the second round would be a tie, the third round would go to the one who wasn't chosen to break the tie, the fourth to A, the fifth a tie, and so on.

@k98kurz I think the issues that lead to repeal, were related to it's use with small district magnitude, not any attributes of cumulative voting itself. With two seats per district and everything else discouraging the establishment of a third party, the seats end up evenly divided between the two parties unless there is a huge difference in the amount of votes each receives.

You have a nice description of how to add a automatic runoff to a multi-winner election. However, what your describing would be a type of Block Star, it is not a type of proportional representation. Your weighting step achieves a different sort of proportionality then what is referred to as proportional representation.

@cfrank @Jack-Waugh Drumming short slogans without clear definition or explanation can be problematic. Even "one ballot per voter" could lead to people insisting that all offices and questions, up for vote on the same day being crammed on one sheet of paper. Or originalist judges going back to little ball used for secret voting as the definition of ballot. We need to remind people that a vote: is the opinion of one member of a group, used in an effort to determine the opinion of the group, on a matter that they are trying to make a decision upon. The more often people are exposed to longer explanations of what a vote is, the less they will assume a narrow definition based on exposer to only one voting method.

@k98kurz I think the issues that lead to repeal, were related to it's use with small district magnitude, not any attributes of cumulative voting itself. With two seats per district and everything else discouraging the establishment of a third party, the seats end up evenly divided between the two parties unless there is a huge difference in the amount of votes each receives.

@gregw Legislation could be crafted that establishes Score Voting as the preferred method and Approval Voting as the default, but that will cast doubt on the constitutionality of score voting and make it easy for courts to reject. Instead I'd suggest splitting score voting into two separate pieces of legislation. The first would establish the score ballot format, and rules for filling it out. The second would cover tabulating score ballots and determining the winner.
No state constitutions have any language that set the ballot format, set rules for filling it out, or have any restrictions preventing the use of a more expression ballot. If would be difficult for the court to fabricate any justification for tossing out the ballot format. The vast majority of state constitutions contain language on tabulating ballots and determining the winner, and that was written without consideration of other alternate voting methods so it's awkward to apply to those situations. So it would be easy for the court the reject the second piece on tabulating ballots on some perceived conflict.If the court keeps the first and throws out the second they stick themselves with the task of interpreting how to apply the existing legislation for tabulating choose-one ballots to the task tabulating score ballots. The language doesn't fit the situation, so multiple interpretations are possible, but every one I've tried results in either score voting or approval voting. A judge faced with such a choice would be likely to just let score voting stand in its entirety.

@sander, sequential bottom two runoff, is a Smith efficient Condorcet procedure. It will always select a member of the Smith set and the Condorcet winner if there is one. Though the procedure you describe is different, it will produce the same results as Smith//Score. With score relegated to the relatively trivial role of cycle breaking, properties of Condorcet systems will predominate. By contrast the top two runoff of STAR will block a Condorcet loser, but a Condorcet winner can still lose if they don't get a good enough score to get in the runoff. This is enough for STAR to retain some significant properties of Score.

@wolftune

The above examples illustrate two issues. Keep in mind that the goal of some supporters of cardinal PR is not to satisfy a strong form of Proportionality for Solid Coalitions (PSC); but to elect one moderate centrist, and a representative for each solid coalition with a quota of supporters. This should often be an achievable goal as the number of candidates entitled to a seat under PSC is usually less than the number of seats to be filled. But by starting off with awarding a seat to the overall score winner electing a moderate is prioritized over PSC.

A second issue, seen in the second example, is that a candidate receiving only low scores can beat a candidate that is clearly entitled to a seat under PSC. This is due to a downward shift in what constitutes a neutral score as the number of seats increases. What I mean by neutral score is the minimum score that a candidate must exceed to be entitled to a seat if everyone bullet votes. This would be the corresponding score to the vote exceeded in the Droop Quota. For 0 to 5 score this would be 2.5 for a single winner election, 1 ⅔ for two winners, 1.25 for three, 1 for four, etc. For a six seat election in the above example a score of 1 isn't a low score of disapproval it's a modest score of approval.

What could be done about these issues?

1: Educate voters on strategy.
Allocated Score will achieve a proportional outcome, if all voters engage in bullet voting. Rating all candidates that the voter doesn't approve of with a score of zero is essential. The ballot options could be changed from (0, 1, 2, 3, 4, 5) to (0, 0, 0, 3, 4, 5) to stress the point. The description of STAR-PR doesn't have the top-two runoff step of STAR so there's no incentive to use intermediate scores as it is.

2: Award seats to candidates that qualify for seats based on their top scores alone first. For example: count only the ratings of 5 cast then seat the candidates that have a score of 5 times the Droop quota, following a procedure similar to the first round of PR-STV. Afterwards this would switch to Allocated Score instead of the elimination round. This should prioritize proportionality, while preserving some of the attributes of Allocated Score. Intermediary rounds could be included, counting scores of four and five and a round counting scores of three, and five in a Bucklin like manner. This would ensure that even parties that split their vote between multiple candidates would get priority over an overall score winner. There would be a confirmation process to handle surplus votes and make sure votes aren't counted twice, that would seat individual candidates sequentially, but this would fundamentally change the process from a series of single winner elections to rounds of multi-winner elections. Multiple candidates could meet the Droop quota to win the round, but the ballot deweighting process could result in not all winners being confirmed.

3: Transform the scores to adjust for the shift in the neutral point.
By adjusting the scores before they are tallied voters can score on the more familiar scale used for single winner elections and have them corrected for the distortion caused by having multiple winners.
Example: four winner election 0 to 5 ballot. The neutral score would be 1. The balance test for the equal vote criteria also changes for multi-winner. Instead of the familiar one vote canceling one vote, four votes in opposition are necessary to cancel each vote in favor(see balance test below). A score of zero remains zero, 1 becomes ⅓, 2 becomes ⅔, 3 becomes 2 ⅔, 4 becomes 3 ⅔, and 5 remains 5. Consider if four voters give a candidate a zero and one a five. Total score will be 5 and the average score will be 1. If four give a 1 and one a 4. Total score would be 4 * ⅓ + 3 ⅔ = 5 and an average of 1. If four give a score of 2 and one a 3. Total score would be 4 * ⅔ + 2 ⅓ = 5 and an average of 1. With these adjustments in place the potential harm to a preferred candidate of giving a lesser candidate an intermediate score will be on par with single winner systems.

Balance Test for Score (Based on Droop Quota rather than majority, to be applicable to any number of winners) : There shall be a set of ballots of size one plus the number of seats to be filled, such that if one of the ballots is cast any way possible the other ballots can be cast to cancel out the vote leaving the average scores for each candidate unchanged. The ballots shall be scored such that, scores of approval on one of the ballots shall be countered by the opposing score of disapproval on the other ballots, and a score of disapproval shall be countered by the opposing score of approval on one of the other ballots and the score of disapproval shall be shared with the remaining ballots.

4:Only count a fraction of the votes for each candidate.
Sequential Monroe avoids the issues by only counting the top Hare quota of ballots for each candidate. With the vast majority of low scores left uncounted, the results are largely unaffected by the distortion of scores in multi-winner elections. However, with little more than the candidates core supporters contributing, the results will lean less moderate until the final round.

What if STAR-PR was not a misleading name; and included an automatic top-two runoff based on all ballot preferences before each selection was finalized?

With the current version of Allocated Score if all voters make use of intermediate scores it will destroy any semblance of proportionality. If all voters resort to min-max voting to protect proportionality the runoff will be irrelevant. More likely voters most interested in getting a particular candidate or party elected or dead set against a particular candidate, will resort to min-max voting while voters torn between objectives will make use of the full range of scores. Vote splitting among min-max voters could result in parties failing to get a seat, and the voters making full use of all scores will have an outsized influence in getting their lesser evils elected.

Score based PR systems have the potential to provide a balance between different interests. With some of the above changes in place, parties with a quota of supporters should get one of their top two candidates elected, even if they split their votes between multiple candidates, but could end up with the one preferred by voters overall, rather than the one preferred by party members.

@toby-pereira I came up with 67 seats for A and 33 seats for B, but that was based on the example in the Wikipedia wiki not Electowiki. That example suggests a cumulative cost tally for each party that is added on to after each seat is apportioned. Party A would have a starting cost of 1/3 or 2/6 and B would be 1/2 or 3/6. When A is selected 1/3 or 2/6 is added to A's cost and (1/2)/3 or 1/6 is added to B's cost to reflect the fraction of B's voters who also voted for A. When B is selected 1/2 or 3/6 is added to B's cost and (1/3)/2 or 1/6 is added to A's. Each round goes to the candidate with the lowest cumulative cost. So the first round would go to A, the second round would be a tie, the third round would go to the one who wasn't chosen to break the tie, the fourth to A, the fifth a tie, and so on.

@denis-falvey I read the above preprint. Your Legislative Empowerment Measure (LEM) seemed clear and logical to me, but I didn't scrutinize every detail. I had almost no prior familiarity with Gini coefficients or Lorenz curves before, but good explanations were easy to look up. I could see how the same concept could be used for malapportionment; but with no axes labels on your Lorenz curve graphs I still couldn't follow what you had done.

The description of SMDPR (single member district proportional representation) in the above preprint lacked details necessary to assess if the numbers you extrapolated were even plausible. I did, however, find a detailed description at APSA Preprints. The negligible surplus vote looks suspicious, but given the use of Hare quota and proportion of vote that year to very small parties, it does seem plausible.

The description of ALRM (accountable local representation measure) has a typographical omission. It is clear that ALRM was calculated as LEM per representative per district not LEM per representative. The bigger issue is that, as a function of districts per representative, ALRM is only useful for comparing election systems for the same size country with the same number of representatives. A universal measure could be a function of districts per citizen, but that would make it an accountable personal representation measure, a very small number. Or it could simply be a function of districts, or districts per thousandths of the country's citizens, to bring it back to a similarly scaled number.

I also question the utility of a combined measure when the subject matter could be conveyed with greater clarity using independent threshold and limit measures, that would answer the following questions.

If all qualified citizens cast valid ballots, what would be the minimum number of votes needed by an individual candidate to clinch a seat? What percentage of votes in the riding would this be? What percentage of votes in the allocation division would this be? What percentage of votes in the country would this be? If the choice of riding would yield different results express the answer as a range.

What would be the maximum number of votes that could be combined toward selecting an individual candidate?(same conditions and supplemental questions for this and the following)

What would be the minimum number of votes needed to clinch a seat for a party if their supporters are optimally concentrated?

What would be the minimum number of votes needed to clinch a seat for a party if their supporters are evenly distributed throughout the country?

What would be the maximum number of votes that can be combined toward securing seats for a party?

@sarawolk did you check spam folders? When I set up an account, Gmail auto-filed the confirmation email as spam.